Real and complex Monge-Ampère equations, statistical mechanics and canonical metrics

Abstract: Recent decades has seen a strong trend in complex geometry to study canonical metrics and the way they relate to geometric analysis, algebraic geometry and probability theory. This thesis consists of four papers each contributing to this field. The first paper sets up a probabilistic framework for real Monge-Ampère equations on tori. We show that solutions to a large class of real Monge-Ampère equations arise as the many particle limit of certain permanental point processes. The framework can be seen as a real, compact analog of the probabilistic framework for Kähler-Einstein metrics on Kähler manifolds. The second paper introduces a variational approach in terms of optimal transport to real Monge-Ampère equations on compact Hessian manifolds. This is applied to prove existence and uniqueness results for various types of canonical Hessian metrics. The results can, on one hand, be seen as a first step towards a probabilistic approach to canonical metrics on Hessian manifolds and, on the other hand, as a remark on the Gross-Wilson and Kontsevich-Soibelmann conjectures in Mirror symmetry. The third paper introduces a new type of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, that generalises Kähler-Einstein metrics. Existence and uniqueness theorems are given as well as a proof of one direction of a generalised Yau-Tian-Donaldson conjecture, establishing a connection between this new notion of canonical metrics and stability in algebraic geometry. The fourth paper gives a necessary and sufficient condition for existence of coupled Kähler-Einstein metrics on toric manifolds in terms of a collection of associated polytopes, proving this generalised Yau-Tian-Donaldson conjecture in the toric setting.